In 1972, Gudzenko and Yakovlenko.sup.1 described a process involving, effectively a three-body collision between atomic or molecular species X and Y, and a photon h.omega.. FIG. 1a shows the energy level for atoms X and Y that corresponds to EQU X(1)+Y(2)+h.omega..fwdarw.X(2)+Y(1) (1).
The absorption of the photon, having an energy h.omega., allows a resonant two-body collision.
The production rate of state X(2) can be written ##EQU2## where [] indicates concentration, .rho. is the photon flux field and k is a three-body rate coefficient for the radiative collision. If we look at k.rho. in terms of a normal binary collision, then EQU k.rho.=&lt;.sigma.v&gt;, (3)
where .sigma. is the event cross section and v is the velocity, thus (2) becomes ##EQU3## Alternatively, if we look at k in terms of photon absorption, then EQU k[Y(2)]=B.sub.12, (5)
where B.sub.12 is an Einstein-like absorption coefficient. Now (2) can be written as ##EQU4## When cast in the form of a collision, as in (3) and (4), the cross section .sigma., becomes a function of .rho., the photon flux field. When written in the form of a radiative absorption, as in (5) and (6), the Einstein stimulated absorption coefficient, B.sub.12, becomes a function of the density, [Y(2)]. The photon, h.omega. does not have the energy of the difference between X(2) and X(1), but approximately the energy difference between X(2) and Y(2). A third method of describing these collisions would be the absorption of a photon by a quasi-molecule or collision complex Y(2)X(1). This model is conceptually useful.
Harris.sup.2 has used the collisional model to describe his observations of many such collisions that he and his colleagues have observed. In this work very large cross sections have been observed for a wide variety of collisions induced by intense photon flux fields. The largest cross section reported thus far is 8.times.10.sup.-13 cm.sup.2..sup.2 Harris.sup.4 and others.sup.5 have used these large cross sections to propose population mechanisms for lasers.
We propose a stimulated emission process based on the inverse process to (1). This process was considered briefly in reference 1. EQU h.omega.+X(2)+Y(1).fwdarw.X(1)+Y(2)+2h.omega. (7).
For this reaction, the colliding atoms are stimulated to emit a photon, (FIG. 1b), where in reaction (1) the colliding atoms are stimulated to absorb a photon. The cross section for these two reactions, (1) and (7), are the same; thus the rate of production will be the difference between the two processes ##STR1## The photon production rate ##EQU5## is the negative part of (8). If the statistical weights are included, then the gain in photon flux can be expressed as ##EQU6## Note that the gain of such a system depends on an inversion of the products of the densities. This provides new ways to produce inversions, for the energy storage can be in one species for the upper quasi-molecular laser level and the lower level can be deactivated in the collision partner. At high photon flux fields, a large cross section typical of Harris' data.sup.2 would force the radiative collision to be the chief energy pathway, making the photon production efficiency approach the quantum efficiency. At lower photon flux fields, the energy is channeled through other processes and the efficiency would be expected to be very low. As an example helium and nitrogen are chosen as the media. Although better systems may exist, the abundance of atomic and molecular data for helium and nitrogen makes this example useful.
The energy level diagram for helium and nitrogen in FIG. 2 can be correlated to that of FIG. 1 by, ##EQU7## Note that the * denotes excitation to a Rydberg state near the ionization threshold of nitrogen. This depresses the energy of the B core, v=4, N.sub.2 state to resonance with He(2.sup.3 S). The reaction of interest now becomes ##EQU8##
Because the Rydberg electron is near the ionization limit (in or near the Saha region), the Y(3), N.sub.2 *(B,v=4), state is resonant with He(2.sup.3 S). Also, the induced transition can be considered in the same manner as the equivalent ionic state. The Franck-Condon factor.sup.7 shows that the transition 3538 .ANG.is optimal for reaction (11).
Harris.sup.3 has derived an expression for the cross section of a dipole-quadrupole radiative collision for both strong and weak photon flux field regimes. Using this expression for the helium-nitrogen system, we get ##EQU9## The dipole matrix element, .mu..sub.21, (N.sub.2 .fwdarw.N.sub.2 *(B,v)), is determined by the energy.sup.6 and the Franck-Condon factor.sup.7. The matrix element, .mu..sub.23, (N.sub.2 *(B,v).fwdarw.N.sub.2 *(x, v-1)), is weighted by the Franck-Condon factor.sup.7. The quadrupole matrix element, q.sub.12, (He(1.sup.3 S).fwdarw.He(2.sup.3 S)), is estimated from the equivalent singlet lifetime assuming electron exchange during the collision. The Weisskopf radius.sup.8 is .rho..sub.0, .DELTA..omega. is the detuning energy (normally the energy difference between the virtual state and the real state), v is the thermal velocity, and E is the electric field due to the photon flux.
Since the Rydberg state is effectively in the Saha continuum, the detuning energy, .DELTA..omega., is taken to be a collection of the linewidths of the three states and the bandwidth of the incoming photon flux field. The value of the detuning energy is estimated to be 2 cm.sup.-1. FIG. 3 shows the limiting case cross sections calculated for (11) as a function of photon flux field, for both the strong and weak photon field cases. A model has been developed for a helium-nitrogen system using the cross section shown in FIG. 3.
The rate equations used in this system involve the concentrations of He.sup.+, He.sub.2.sup.+, He(2.sup.3 S), and N.sub.2.sup.+.
The rate of He production is determined by the source terms involving S, the power deposition and the W value or energy investment per ion, and metastable-metastable ionization, He(2.sup.3 S)+He(2.sup.3 S).fwdarw.He.sup.+ +He+e. The loss terms involve charge exchange, N.sub.2 +He.sup.+ .fwdarw.N.sub.2.sup.+ He, and three-body conversion, 2He+He.sup.+ .fwdarw.He.sub.2.sup.+ +He. The equation for He.sup.+ production is ##EQU10## The coefficients are listed in Table 1.
The He.sub.2.sup.+ rate equation is given by ##EQU11## The single source term is three-body conversion, while the loss terms involve collisional radiative recombination, He.sub.2.sup.+ +e+x.fwdarw.CRR, and two- and three-body charge transfer, N.sub.2 +He.sub.2.sup.+ .fwdarw.N.sub.2.sup.+ +2He and N.sub.2 +He.sub.2.sup.+ +He.fwdarw.N.sub.2.sup.+ +3He. See Table 1 for the coefficients of this rate equation.
The metastable production rate has two source terms, one dependent on the energy deposition and the other dependent on collisional radiative recombination. The loss terms depend on metastable-metastable ionization, two- and three-body Penning ionization, N.sub.2 +He(2.sup.3 S).fwdarw.N.sub.2.sup.+ +He+e and N.sub.2 +He(2.sup.3 S)+He.fwdarw.N.sub.2.sup.+ +2He+e, and super=elastic relaxation, He(2.sup.3 S)+e.fwdarw.He+e (20 eV). Other losses involve three-body conversion to molecular metastable, He(2.sup.3 S)+2He.fwdarw.He.sub.2 (2.sup.3 .SIGMA.)+He, spontaneous emission of a photon by a radiative collision, N.sub.2 +He(2.sup.3 S).fwdarw.N.sub.2 +He+h.omega., and stimulated emission from a radiative collision, N.sub.2 +He(2.sup.3 S)+h.omega..fwdarw.N.sub.2 (x,v)+He+2h.omega.. The rate equation for He(2.sup.3 S) is ##EQU12##
TABLE 1 ______________________________________ T.sub.e = electron temperature, T.sub.o = plasma temperature, n.sub.e = electron density, n.sub.o = neutral density, p.sub.He = partial pressure of helium. RATE COEFFICIENT VALUE USED REFERENCE ______________________________________ .beta. 1.8 .times. 10.sup.-9 cm.sup.3 /sec 9 k.sub.1 1.2 .times. 10.sup.-9 cm.sup.3 /sec 10 k.sub.2 67.0 .+-. 5 Torr.sup.-2 /sec 11,12 .alpha. 4.5 .times. 10.sup.-20 (T.sub.e /T.sub.o).sup.-4 n.sub.e 12 + 5.0 .times. 10.sup.-27 (T.sub.e /T.sub.o).sup.-1 n.sub.o cm.sup.3 /sec k.sub.30 1.1 .times. 10.sup.-9 cm.sup.3 /sec 13 k.sub.31 1.6 .times. 10.sup.-29 cm.sup.6 /sec 13 S.sub.m S/0.56 14 k.sub.40 6.9 .times. 10.sup.-11 cm.sup.3 /sec 15 k.sub.41 2.9 .times. 10.sup.-30 cm.sup.6 /sec 15 k.sub.5 7.0 .times. 10.sup.-11 (T.sub. e).sup.1/2 cm.sup.3 /sec 16 .beta..sub.2 0.3 Torr.sup.-2 /sec 17 A' 8.pi.h.nu..sup.3 .sigma.'-v/.rho.c.sup.2 Einstein Coefficient -v 2.5 .times. 10.sup.5 cm/sec Thermal Velocity .alpha..sub.N.sbsb.2.spsb.+ 2.2 .times. 10.sup.-7 cm.sup.3 /sec 18 ______________________________________ Coefficients for equation (16) are also defined in Table 1.
The ionized molecular nitrogen rate equation is ##EQU13## All the source terms have been defined previously and the loss term involves dissociative recombination, N.sub.2.sup.+ +e.fwdarw.N.sub.2 *+N. The rate coefficients are given in Table 1.
Finally, a charge balance equation is used to conserve the system's charge.
These equations have been solved in steady state for power depositions between 200 W/cm.sup.3 and 2 MW/cm.sup.3 in a mixture of one atmosphere of helium and various percentages of nitrogen. The data presented here is for a mixture with 1% nitrogen.
The gain for the preceding system is calculated from equation (10). It should be emphasized that the N.sub.2 *(x,v=3) state is autoionizing.sup.19 and has a lifetime of about 10.sup.-10 seconds. This makes the product density, [He(1.sup.1 S)][N.sub.2 *(x,v=3)], negligible since the lower levels self-destruct. The calculated gain is shown in FIG. 4. The decrease in gain at the higher photon flux fields is due to the high destruction rate of helium metastables, the effects of which are shown in FIG. 5. The calculated efficiency (ratio of radiative power to power deposition) is shown in FIG. 6 and saturates near the quantum efficiency of 15%.
It should be emphasized that, although the gain for the system is large (see FIG. 4), significant energy loss due to superradiance will not be a problem due to the low efficiency at small photon flux fields. Significant energy extraction will not only occur in the direction of the incoming oscillator beam, since the intensity of the photon field determines the cross section for radiative collisions.
Further analysis and experimental results are presented in the section on carrying out the invention, and in FIGS. 8-16.